this reduction may be made by calculation, or by reference to tables of reduction usually engraved on the vertical arcs of angular instruments, which, while they show on one side the angle of elevation or depression, give on the other the number of units per hundred that have to be deducted to reduce the hypotenuse to its corresponding horizontal length. This subject I do not at present wish to discuss, especially as in small surveys, performed with the chain only, an allowance or reduction is generally made in the field by construction or estimation as the measurement proceeds.

The Reduction may be effected by Construction.

If the slope be not very steep, the reduction is accomplished by holding the lower end of the chain above the ground, as nearly horizontal as can be judged by the eye, allowing a pointed plummet to hang from the hand that holds the chain, to indicate where the arrow shall be placed. If the slope be steep, one half or one quarter of the chain is raised, as being more easily brought to a horizontal position; and on precipitous banks the offsetstaff or measuring tape is substituted, as giving more correct results, with greater expedition. It may be observed, that when the chain is thus held suspended, it cannot be straightened, its links describing the catenary curve; but as a compensation for the shortening of the chain caused by the bend, it is found that the pull at each end of the chain to diminish the curvature caused by its weight, tends to open the unwelded elastic rings, and thus to add very sensibly to the length which it would have when laid upon the ground. Nevertheless, the bending of the chain is an element of inaccuracy in this process, which is further made erroneous by the difficulty of ascertaining exactly without lateral or longitudinal error, the point vertically

beneath the elevated extremity of the chain; and by the unavoidable deviation from the horizontal line, when it has to be estimated by the eye.

The Reduction may be effected by Estimation.

The surveyor who does not use an angular instrument for the purpose of ascertaining the required reduction, learns by habit to estimate and make at each chain's length on the ground an approximate reduction. In such an operation he will be much assisted by the subjoined table, which may be copied on the first leaf of the field-book, and the principal elements of which are easily learned after a few references and practical applications. The inclination in such cases is, of course, estimated solely by the eye; and I describe this method, not to recommend it, but because it will aid the surveyor in approaching to accuracy, in exceptional instances, where he may not have the assistance of an angular instrument. The reductions are purposely made approximate in the table, in order not to distract the attention by fractional quantities in the application of a process, in itself only an approximation.

REDUCTION in links upon each length of 100 links for the following Angles of Inclination, or for the following Rates of Inclination expressed in the terms of the Horizontal Base and Vertical Height.

The necessary reduction, as estimated on the ground, is effected as the measurement proceeds, by putting the chain forward the exact number of links denoted by the table as due to the angle, or the rate of inclination. This mechanical method possesses this advantage, that the crossing of the fences or natural boundaries, and the position of the offsets, are at once entered in the field-book, with the required reduction. Practised surveyors obtain by this simple means, results much more accurate than would have been expected, but I repeat, when perfect accuracy is sought, and when the survey is extensive, the angles of inclination should be observed, and the proper deduction obtained by computation, and allowed when the work is being plotted. This part of the subject is fully explained in treating of levelling and the interior filling-in of a trigonometrical survey.

PRACTICAL METHODS OF MEASURING INACCESSIBLE DISTANCES, AND OF AVOIDING OBSTACLES IN RUNNING LINES.

Cases of obstruction in the measurement of a line offered by the intervention of trees, buildings, rivers, lakes, &c., are readily overcome by practical geometry, even without the aid of an angular instrument. But when the difficulty cannot be surmounted with ease by the chain, it is always better to make use of some angular instrument, which, with the aid of plane trigonometry, will enable the surveyor to solve all difficulties.

In ranging his lines, the surveyor should be careful to dispose them so that they shall, if possible, pass clear of trees, houses, and other impediments. If, in spite of all his efforts to the contrary, he finds it impracticable to avoid them altogether, he may proceed thus for passing them.

The measured line A B being obstructed by a tree, a brook, or a house, &c., staves are set up at C, D, and E, at equal distances from the measured line, and far enough from it to enable the new line C D E F, to pass clear of the obstacle. This new line, parallel to AB, is then measured till the obstruction is passed, when by setting other staves, G, H, I, at distances from the second line equal to those first set out, a return is made to the direction of the original line, which is pursued as before.

When the obstacle thus avoided is a tree, it is called a "sight tree," and for the purpose of facilitating future reference, it is marked in a particular manner, by an arrow-head or otherwise, cut at or near the points where the direction of the line meets the tree both in front and rear.

Another method of passing such obstacles is by the construction of equal triangles: thus, let A D be the direction of the line under measurement, the further progress of which is interrupted at A. From A measure A C in any direction, and leave a central mark B; from D, an acces

sible point on A D, measure a line D B E, making B E equal to B D, then CE will be equal to AD, the distance required. This method is inapplicable if the line has not been previously ranged and determined by signals fixed beyond D, and visible from it.

The following method, embodying the same property of equal triangles, may also be adopted, under different circumstances. At A and D erect staves; measure a line A C, at any angle with A D, and leave a central mark B. Measure a second line F BG, making B G equal to FB,

join G C and produce it to E, the point of intersection of the lines D B and G C; CE will be equal to A D, the distance required. In this case also, points ranged in the continuation of A D, must be visible from D.

To avoid a similar obstacle, using an angular instrument. The measured line A B is interrupted at B; make

angle B C D equal to one-third of two right angles, and proceed along C D to a point D, making CD equal to B C; at D, measure the angle CDE equal to twothirds of two right angles. The triangle B C D is by construction equilateral, each angle being equal to one-third of two right angles; hence BD is equal to B C or CD; and D E is continued in the direction of the original line.

Required along the line A B produced, the distance B O, inaccessible to direct measurement with the chain. At B raise the perpendicular B C of any convenient length, by making with the chain a triangle whose sides are in the ratio of 3, 4, and 5, (see page 10). At C, range in